Stat 427/527: Homework #5 and #6 due Tuesday, November 1, 2005 *************************** This HW is worth 20 points. *************************** 1. The following determinations of the parallax of the sun (the angle spanned by the earth's radius as if it were viewed and measured from the sun's surface) were made in 1761 by noted astronomer James Short. The units are in seconds of a degree (1/360 degree). 8.50 8.06 8.65 9.71 8.80 7.99 8.50 8.43 8.35 8.50 8.40 8.58 7.33 8.44 8.71 8.28 8.82 8.34 8.64 8.14 8.31 9.87 9.02 9.64 9.27 7.68 8.36 8.86 10.57 8.34 9.06 10.34 8.58 5.76 9.11 8.55 9.25 8.07 7.80 8.44 8.66 9.54 9.09 8.36 7.71 8.23 8.34 9.07 8.50 9.71 8.30 8.50 8.60 With a careful determination of the radius of the earth and a good average value of the parallax, the average distance of the earth to the sun can be obtained. The currently accepted value of the parallax is 8.798. Within this framework, define mu = the population mean of all potential measurements of the parallax using Short's device. We are interested in whether mu could be the currently accepted value of the parallax of the sun, that is, we wish to test whether mu = 8.798. a. Describe the distribution of determinations of the parallax. Be complete. b. Perform the standard t-test on these data, at the 5% level, and construct a 95% CI for mu. Interpret the results, given the question of interest. c. Repeat the analysis using a suitable non-parametric method, and contrast the results with part b. Which analysis seems most reasonable, and what are your conclusions based on that analysis, given the question of interest? 2. The data below are the survival times in hours of 72 guinea pigs after they were injected with a given dose of tubercule bacilli in a medical experiment. The data are from the article ``Acquisition of resistance of guinea pigs injected with different doses of virulent tubercule bacilli,'' by T. Bjerkedal in the American Journal of Hygiene, (1960), pp. 130-148. 43 45 53 56 56 57 58 66 67 73 74 79 80 80 81 81 81 82 83 83 84 88 89 91 91 92 92 97 99 99 100 100 101 102 102 102 103 104 107 108 109 113 114 118 121 123 126 128 137 138 139 144 145 147 156 162 174 178 179 184 191 198 211 214 243 249 329 380 403 511 522 598 a. Obtain a 95% t-CI for the mean survival time. b. Repeat part a using a suitable nonparametric method. c. Take the log of survival time and find a 95% t-CI for mean log survival time. d. Repeat part c using a suitable nonparametric method. e. Compare your 4 CI's, and contrast the nonparametric with the t-CI's. If they differ much, explain why they differ. Which analysis appears most appropriate? Explain. 3. In an 1898 Biology lecture at Woods Hole, Massachusetts, Hermon Bumpus reminded the audience that the process of natural selection for evolutionary change was an unproved theory. As evidence in support of natural selection, he presented measurements on house sparrows brought to his Brown University laboratory after an uncommonly severe winter storm. Some of the birds had died, and some had survived. Bumpus asked whether those that perished did so because they lacked physical characteristics enabling them to withstand the intensity of that particular instance of selective elimination. The data we will look at are the humerus (arm bone) lengths for the 24 adult male sparrows that perished and the 35 adult males that survived. The data are on the web page. You will see that the data are in two columns. The first column contains the humerus lengths for the 59 birds. The second column identifies whether the birds perished (1) or survived (2). a. Make appropriate graphical displays to compare the humerus lengths in the two samples b. Test at the 5% level whether there is any difference in the population mean humerus lengths for those that perished and those that survived. Use both the t-test and an appropriate nonparametric procedure. c. Compute and interpret a 95% CI for the difference in population mean humerus lengths for those that perished and those that survived. Repeat for an appropriate nonparametric procedure. d. Discuss any statistical assumptions that you have made in carrying out the analyses, and whether the assumptions seem reasonable. e. Write a short summary for the problem. What analysis seems most appropriate? 4. Protoporphrin levels were determined for three groups of people - a control group of normal workers, a group of alcoholics with sideroblasts in their bone marrow, and a group of alcoholics without sideroblasts. The given data appeared in the paper "Erythocyte Coproporphyrin and Protoporphrin in Ethanol-Induced Sideroblastic Erythoporiesis" (Blood, 1974, p. 291-295). Analyze the data, assuming you are interested in comparing the typical protoporphrin level across groups. Quantify any differences you find. Make sure to clearly define all population parameters, and assess the assumptions underlying your chosen method of analysis. Normal group: 22 27 47 30 38 78 28 58 72 56 30 39 53 50 36 Alcoholics with sideroblasts: 78 172 286 82 453 513 174 915 84 153 780 Alcoholics without sideroblasts: 37 28 38 45 47 29 34 20 68 12 37 8 76 148 11